WALL-CROSSING INVARIANTS: FROM QUANTUM MECHANICS TO KNOTS
D. GalakhovaK A. Mironova>€>d**, A. Morozova>d***
a ITEP, 117218, Moscow, Russia bNHETC and Department of Physics and Astronomy, Rutgers University, Piscataway
NJ 08855-084 USA ' l.i In ih r Physics Institute, 119991, Moscow, Russia dNational Research Nuclear University MEPhI, 115409, Moscow, Russia
Received October 31, 2014
We offer a pedestrian-level review of the wall-crossing invariants. The story begins from the scattering theory in quantum mechanics where the spectrum reshuffling can be related to permutations of S-matrices. In nontrivial situations, starting from spin chains and matrix models, the S-matrices are operator-valued and their algebra is described in terms of 7v- and mixing (Racah) W-matrices. Then the Kontsevich-Soibelman (KS) invariants are nothing but the standard knot invariants made out of these data within the Reshetikhin-Turaev-Witten approach. The 7v and Racah matrices acquire a relatively universal form in the semiclassical limit, where the basic reshufflings with the change of moduli are those of the Stokes line. Natural from this standpoint are matrices provided by the modular transformations of conformal blocks (with the usual identification 7v = T and U = S), and in the simplest case of the first degenerate field (2.1), when the conformal blocks satisfy a second-order Shrodinger-like equation, the invariants coincide with the Jones (Ar = 2) invariants of the associated knots. Another possibility to construct knot invariants is to realize the cluster coordinates associated with reshufflings of the Stokes lines immediately in terms of check-operators acting on solutions of the Knizhnik-Zamolodchikov equations. Then the 7v-matrices are realized as products of successive mutations in the cluster algebra and are manifestly described in terms of quantum dilogarithms, ultimately leading to the Hikami construction of knot invariants.
Contribution for the JETP special issue in honor of V. A. Rubakov's 60th birthday
Contents
1. Introduction..................................................................................................................................................................624
2. Wall crossing formulas as a piece of the WKB theory....................................................................626
2.1. Asymptotic behavior......................................................................................................................................................626
2.2. .Jumps of WKB network topology on the curve..................................................................................................628
2.3. Nontrivial moduli space invariants: wall-crossing formulas in the moduli space....................................629
3. Classic problem of quantum mechanics: double-well potential..................................................630
4. Check-operator "quantum, refined"..............................................................................................................632
4.1. Intuitive remarks..............................................................................................................................................................632
4.2. Beta-ensemble construction........................................................................................................................................633
* E-mail: galakhov'flitep.ru; galaldiov'fflphysics.rutgers.edu
E-mail: mironov'ffllpi.ru
E-mail: morozov'flitep.ru
4.3. Determinant check-operator: quantizing the spectral curve..........................................................................634
4.4. Higher weight operators and spectral covers........................................................................................................637
5. Knot invariants from WKB morphisms....................................................................................................639
5.1. Reidemeister invariants from quantum field theory..........................................................................................639
5.2. Knot invariants from the RTW representation via duality kernels..............................................................641
5.2.1. The basic idea..................................................................................................................................................................641
5.2.2. 5 in (5.19) as the Racah matrix................................................................................................................................642
5.2.3. S and T matrices from conformai theory..............................................................................................................643
5.2.4. Plat representation of link diagrams (spherical conformai block)................................................................645
5.3. Hikami knot invariants from check-operators......................................................................................................649
5.3.1. Quantum spectral curve in Chern Simons theory..............................................................................................649
5.3.2. Verlinde operators..........................................................................................................................................................650
5.3.3. Knots and flips..................................................................................................................................................................651
5.3.4 Hikami invariant as a KS invariant..........................................................................................................................652
5.4. Stokes phenomenon in conformai blocks................................................................................................................653
5.4.1. WKB approximation......................................................................................................................................................653
6. Conclusion and discussion..................................................................................................................................653
Appendix..........................................................................................................................................................................657
A.l The special case q —¥ 1..................................................................................................................................................657
A.2 Explicit calculations for q ^ 1....................................................................................................................................660
References........................................................................................................................................................................660
DOI: 10.7868/S0044451015030234
1. INTRODUCTION
The string theory approach to any problem is to consider it together with all possible deformations and as a particular representation of some general structure appearing in many other, seemingly unrelated problems in other fields of science. One of the fresh applications of this approach is the study of the wall-crossing phenomena (phase transitions) and associated invariants, which remain the same after reshuffling. The outcome of this study is that the Kontsevich Soibelman (KS) invariants fl] found so far on this way, are probably not that new: they belong to an old class of invariants of the Reshetikhin Turaev Witten type, of which the best known are knot invariants [2, 3]. At the same time, what naturally arises in wall-crossing problems is quantum /^.-matrices in representations less trivial than the Verma modules of SUq(N), and this can further stimulate the study of knot invariants in nontrivial representations.
An archetypical example of the wall crossing is the spectrum dependence on the scattering potential in quantum mechanics. We consider a particle in the infinite well with some localized potential, for example:
(^a;2+ a-2)</•(.<:) = o, v(-Li) = hl-2) = 0. (i.i)
The spectrum k(u) is defined by the spectral equation
sin (kLi + kL-2^ — j sm(kLi) sm(kL-2) = 0 (1.2)
and changes from the set
kn = m>' at u =0 (1.3)
L<i + L'2
to the union of two sets1)
11 For u > (Li + Laj/LiLi, there are also two bound states with k = ±in, where k solves the equation
sh (kLi + kLi'] — — sIi(kLi) s1i(kL2) = 0.
k, 1/L 4
___X
3-
1- r
-80 -60 -40 -20 0 -1 20 40 60 80 u, 1/L
-3.
Fig. 1. The picture of energy levels as a function of u at Li = 2L? := 2L
, t 7tn , t r 7Tn /
kn = —, kn = — at it = oc. (1.4)
L\ L'2
The smooth evolution with u is shown in Fig. 1, but the net result is the rather radical reshuffling of (1.3) into (1.4). The task can be to study this reshuffling and to ask if there are quantities that remain the same after the reshuffling.
The question is actually uninvestigated, but its more sophisticated versions were studied and some invariants were revealed (although it is unclear if they reduce to triviality in this original problem).
The point is that there is nothing special about the ¿-function potential: the pattern remains the same for an arbitrary barrier vanishing in the vicinity of the box walls. Any such problem is described in terms of the 2x2 scattering matrix
5: e±ikr\ . a±(k)e±ikx + d±(k)e*ikx\ T . (1.5)
la1 near — Li ^v ' , / near ¿2 v
The spectral equation states that S converts sin (^k(x + ~ eik(x+Li) _ e-%k{x+Li) jnto sjn _ ^ ^ Rik{x-Ln) _ p-lk{x-Ln) ^ por tj10 ¿>_function potential, the scattering matrix is just
S =
( 1-— -— \
2ik 2ik
11 11
V 2ik +2ik )
(1.6)
For the two isolated barriers, the scattering matrix is a product
eikh2 0 0
S! o s2 = • ( „ -S2. (1.7)
where l12 is the distance between the two, and so on: in general, we have the pro
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